ENERGY-STABLE AND MESH-PRESERVING PARAMETRIC FEM FOR MEAN CURVATURE FLOW OF SURFACES\ast

Harmonic maps serve as reliable approximations to conformal maps. Building upon this concept, we introduce a family of novel parametric finite element schemes for solving the mean curvature flow of surfaces in this paper. The key idea involves coupling the normal component of the original equation with a modified harmonic map heat flow. This heat flow induces a map from a given reference surface to the unknown surface to be solved, resulting in a new system that effectively preserves the mesh quality. We employ the linearized Euler scheme and the BDF2 scheme in the temporal direction, with the existence and uniqueness of solutions being rigorously proven. We prove that the Euler scheme is energy-stable, and it becomes energy-diminishing when the current obtained numerical surface is selected as the reference surface for computing the numerical solution at the the advantages of our approach in preserving mesh quality and capturing the evolution of surfaces when they approach a singularity.